# On Sets Defining Few Ordinary Lines

@article{Green2013OnSD,
title={On Sets Defining Few Ordinary Lines},
author={Ben Green and Terence Tao},
journal={Discrete \& Computational Geometry},
year={2013},
volume={50},
pages={409-468}
}
• Published 23 August 2012
• Mathematics
• Discrete & Computational Geometry
Let $$P$$P be a set of $$n$$n points in the plane, not all on a line. We show that if $$n$$n is large then there are at least $$n/2$$n/2ordinary lines, that is to say lines passing through exactly two points of $$P$$P. This confirms, for large $$n$$n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $$n-C$$n-C ordinary lines for some absolute constant $$C$$C. We also solve, for large $$n$$n, the “orchard…
102 Citations
A new progress on Weak Dirac conjecture
• Mathematics
• 2016
In 2014, Payne-Wood proved that every non-collinear set $P$ of $n$ points in the Euclidean plane contains a point in at least $\dfrac{n}{37}$ lines determined by $P.$ This is a remarkable answer for
On sets of points with few ordinary hyperplanes
Let $S$ be a set of $n$ points in the projective $d$-dimensional real space $\mathbb{RP}^d$ such that not all points of $S$ are contained in a single hyperplane and such that any subset of $d$ points
A finite version of the Kakeya problem
• Mathematics
Australas. J Comb.
• 2016
Following Dvir's proof of the finite field Kakeya conjecture and the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, it is proved that a lower bound on the size of S is dependent on the ideal generated by the homogeneous polynomials vanishing on D.
On sets of n points in general position that determine lines that can be pierced by n points
• Mathematics
Discret. Comput. Geom.
• 2020
It is shown that P, the set of points disjoint from P, must be contained in a cubic curve.
Many Collinear $$k$$k-Tuples with no $$k+1$$k+1 Collinear Points
• Mathematics, Computer Science
Discret. Comput. Geom.
• 2013
This work significantly improves the previously best known lower bound for the largest number of collinear k-tuples in a planar point sets, and gets reasonably close to the trivial upper bound O(n^2)$$O(n2). Orchards in elliptic curves over finite fields • Mathematics Finite Fields Their Appl. • 2020 A$$t_ktk Inequality for Arrangements of Pseudolines
A new combinatorial inequality is presented which holds if no more than n-3 pseudolines intersect at one point and is unrelated to the Hirzebruch inequality for arrangements of complex lines in the complex projective plane.
On Sets Defining Few Ordinary Solids
• Mathematics
Discret. Comput. Geom.
• 2021
It is proved that if the number of solids incident with exactly four points of $S$ is less than $Kn^3$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $\mathcal{S}$ are contained in the intersection of five linearly independent quadrics.
On the Number of Ordinary Lines Determined by Sets in Complex Space
• Mathematics
SoCG
• 2017
This paper shows that when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, it is shown that there must be a quadratic number of ordinary lines.
On the Number of Ordinary Conics
• Mathematics
SIAM J. Discret. Math.
• 2016
A lower bound on the number of ordinary conics determined by a finite point set in R is proved, based on the group structure of elliptic curves, that shows that the exponent in the bound is best possible.